Let $X$ be a complex projective variety and denote by $\Omega_X^k$ the (coherent) sheaf of algebraic differential $k$-forms on $X$. Via Serre's GAGA, we obtain a sheaf $(\Omega_X^k)^{\text{an}}$ on $X^{\text{an}}$. Does this coincide with $\Omega_{X^{\text{an}}}^k$, the sheaf of holomorphic $k$-forms on the complex manifold $X^{\text{an}}$? If so, why?
I think I can see this in the case that $X$ is smooth, because then $\Omega_X^k$ is locally free, but I'm not sure for the general case.